International Mathematics Research Notices Advance Access published July 18, 2009 Iwanari, I. (2009) “Integral Chow Rings of Toric Stacks,” International Mathematics Research Notices, Article ID rnp110, 17 pages. doi:10.1093/imrn/rnp110

Integral Chow Rings of Toric Stacks Isamu Iwanari Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan Correspondence to be sent to: [email protected]

The goal of this paper is to compute integral Chow rings of toric stacks and prove that they are naturally isomorphic to Stanley–Reisner rings.

Introduction Intersection theory with integer coefficients of Fulton–MacPherson style on smooth algebraic stacks was developed by Totaro, Edidin–Graham, and Kresch [5, 14, 18]. In particular, the integral Chow ring of a smooth stack which has stratifications by quotient stacks (for example, quotient stacks) was defined. The integral Chow ring of a smooth algebraic stack is an interesting and deep invariant of the stack which reflects the geometric structure together with the stacky structure on it. It is challenging to compute them for interesting algebraic stacks. Some examples of smooth Deligne–Mumford stacks were calculated. For example, in [5], Edidin and Graham calculated the integral Chow rings of the moduli stacks of elliptic curves M1,1 and M1,1 . Vistoli calculated the integral Chow ring of the moduli stack of curves of genus 2 [20]. The purpose of this paper is to compute the integral Chow rings of toric stacks (see Section 1). This paper is motivated by the category-equivalence between the 2category of toric stacks and the 1-category of stacky fans; see [10, Theorem 1.1]. Before Received August 20, 2008; Revised May 7, 2009; Accepted June 22, 2009 Communicated by Prof. Toshiyuki Kobayashi  C The Author 2009. Published by Oxford University Press. All rights reserved. For permissions,

please e-mail: [email protected].

2 I. Iwanari

stating our main result, let us recall the result on Chow rings of toric varieties; see [12], [4, Theorem 10.8], [7, Proposition 2.1]. Theorem (Fulton–Sturmfels, Danilov, Jurkiewicz). Let N = Zd be a lattice. Let  be a nonsingular complete fan in N ⊗Z R, and let X be the associated proper toric variety. Let us denote by A∗ (X ) the integral Chow ring of X . Then there exists a natural isomorphism of graded rings ∼

(Stanley–Reisner ring associated to ) → A∗ (X ). If  is a simplicial fan and the base field is in characteristic zero, then A∗ (X ) ⊗Z Q has a ring structure and there exists a natural isomorphism of graded rings ∼

(Stanley–Reisner ring associated to ) ⊗Z Q → A∗ (X ) ⊗Z Q. (See Definition 2.1 for the definition associated to Stanley–Reisner rings of .) Here we would like to invite the reader’s attention to the fact that if  is simplicial and not nonsingular, the (operational) Chow group Ak (X ) for k ≥ 1 could differ from the module of the “degree k part” of Stanley–Reisner ring of ; see Example 2.12. Furthermore, in such case, a somewhat surprising point is that the Stanley–Reisner ring of  could be nonzero in degrees higher than the dimension of the toric variety X ; see Example 2.12. Since the Chow groups of an algebraic space in degrees higher than its dimension are zero, thus Stanley–Reisner rings give us a combinatorial phenomenon which is unaccountable in the framework of schemes and algebraic spaces. Now we state our main result. Theorem (See Theorem 2.2). Let k be an algebraically closed field of characteristic zero. Let (, 0 ) be a stacky fan and X(,0 ) the toric stack over k associated to (, 0 ). Let A∗ (X(,0 ) ) denote the integral Chow ring of X(,0 ) . Suppose that rays in  span the vector space N ⊗Z R. Then there exists a natural isomorphism of graded rings ∼

(Stanley–Reisner ring associated to (, 0 )) → A∗ (X(,0 ) ). If [V (σ )] and [V (τ )] are torus-invariant substacks in X(,0 ) which correspond to cones σ and τ in , respectively (see Section 2), we have [V (σ )] · [V (τ )] = [V (γ )],

(1)

in A∗ (X(,0 ) ) when σ and τ span γ . (See Definition 2.1 for the definition associated to Stanley–Reisner rings of (, 0 ).)

Integral Chow Rings of Toric Stacks 3

Since the Stanley–Reisner ring associated to a stacky fan (, 0can ) with the canonical free-net 0can coincides with the classical Stanley–Reisner ring associated to  (see Definition 2.1), our result says that there exists an isomorphism of graded rings between the classical Stanley–Reisner ring of a simplicial fan  and A∗ (X(,0can ) ). Here we explain how the usual relations on intersection product of torusinvariant cycles on a simplicial toric variety X (see [6, p. 100]) are derived from that of the toric stack X(,0can ) in A∗ (X(,0can ) ). There exists a coarse moduli map π(,0can ) : X(,0can ) → X . This functor defines the proper pushforward (π(,0can ) )∗ : A∗ (X(,0can ) ) ⊗Z Q → A∗ (X ) ⊗Z Q. By [14, Theorem 2.1.12 (ii)] and [19, Proposition 6.1], (π )∗ induces an isomorphism of groups. Moreover, since a general stabilizer group of a toric stack is trivial, (π )∗ defines an isomorphism of rings [19, (6.7)]. Thus, the ring structure of A∗ (X(,0can ) ) ⊗Z Q yields that of A∗ (X ) ⊗Z Q. The proper pushfor1 ward (π(,0can ) )∗ : A∗ (X(,0can ) ) ⊗Z Q→ A∗ (X ) ⊗Z Q sends [V (σ )] to [V(σ )] since by mult(σ ) [9, Proposition 4.13] the order of stabilizer group of a generic geometric point on V (σ ) is mult(σ ). Here mult(σ ) is the multiplicity of σ , and V(•) is the torus-invariant cycle which corresponds to a cone in . Thus, the relation (1) in A∗ (X(,0can ) ) induces the relations

[V(σ )] · [V(τ )] =

mult(σ ) · mult(τ ) [V(γ )] mult(γ )

in A∗ (X ) ⊗Z Q if σ and τ span γ . From the categorical point of view, the category-equivalence between the 2category of toric stacks and that of stacky fans provides a framework, in which one can naturally understand our main result. In the realm of classical toric geometry, we usually consider the category of nonsingular toric varieties to be a full subcategory of category of (simplicial) toric varieties. In [10, Remark 4.5], it is shown that the category of nonsingular toric varieties can also be naturally embedded into the category of toric stacks as a full subcategory. It seems (at least to the author) that the relation between the Stanley–Reisner rings and toric geometry (Chow rings) is best understood not as a result in the classical schematic toric geometry, but rather as a result that belongs to the geometry of algebraic stacks (in particular, toric stacks). The presented result can be also viewed as an application of intersection theory with integral coefficients [5, 14, 18] to toric stacks. We hope that the reader finds our computation shows a nice relation between toric stacks and combinatorics.

4 I. Iwanari

1 Notation and Preliminaries In this section, we fix notation and prepare for the calculation given in Section 2. Fans. Let N = Zd be a lattice of rank d and M = HomZ (N, Z) the dual lattice. Let •, • be the dual pairing. Let  be a fan in NR = N ⊗Z R (we assume that all fans are finite in this paper). Let (l) be the set of l-dimensional cones. (Thus, (1) denotes the set of rays.) Let vρ denote the first lattice point on a ray ρ ∈ (1). A pair (, 0 ) is called a stacky fan if  is a simplicial fan in NR , and 0 is a subset of || ∩ N, called the free-net of , which has the following property: for any cone σ in , the intersection σ ∩ 0 is a submonoid of σ ∩ N which is isomorphic to Ndim σ such that for any element e ∈ σ ∩ N, there exists a positive integer n such that n · e ∈ 0 . The initial point nρ of ρ ∩ 0 is said to be the generator of 0 on ρ. The positive integer lρ such that lρ · vρ = nρ is said to be the level of 0 on ρ. Notice that 0 is completely determined by the levels of 0 on rays of . Each simplicial fan  has the canonical free-net 0can , whose level on every ray in  is one. Let us give an example of stacky fans. Let σ be a two-dimensional cone in (Z · e1 ⊕ Z · e2 ) ⊗Z R = R⊕2 that is generated by e1 and e1 + 2e2 . Let σ 0 be a free submonoid of σ ∩ (Z · e1 ⊕ Z · e2 ) that is generated by 2e1 and e1 + 2e2 . Note σ 0 ∼ = N2 . Then (σ , σ 0 ) forms a stacky fan. The level of σ 0 on the ray R≥0 · e1 (resp. R≥0 · (e1 + 2e2 )) is 2 (resp. 1). Toric stacks. Let k be an algebraically closed field of characteristic zero. A toric stack over k is a smooth Deligne–Mumford stack X of finite type and separated over k such that (i) there exists an open immersion Gdm → X identifying Gdm with a dense open substack of X , (ii) there exists an action Gdm ×k X → X which is an extension of the action of Gdm on itself, and (iii) the coarse moduli space for X is a scheme. Our definition is taken from [10] (in [10], we called it a toric triple). Given a toric stack X , there exists a stacky fan (, 0 ) such that the associated toric stack X(,0 ) [10, Section 2] or [9] is isomorphic to X . Such a stacky fan is unique up to isomorphism. If ( , 0 ) is a stacky fan in NR with ∼ Zd , then X( , 0 ) is a d-dimensional toric stack (in the above sense) and has a coarse N= moduli map to the toric variety X . For the reader who is familiar with [2], we will remark on a relationship with [2]. More details can be found in [10, Section 5]. In [2], a stacky fan is defined to be a triple (N, , β : Zn → N), where N is a finitely generated abelian group, is simplicial fan in N ⊗Z R consisting of rays {ρ1 , . . . , ρn } = (1), and β : Zn → N is a homomorphism of groups such that the image β(ei ) of N ⊗Z R lies on a lattice point on ρi . Here, ei is the ith standard base of Zn . A stacky fan ( , 0 ) in our sense precisely corresponds to a triple (N, , β : Zn → N) such that N is free and β(ei ) is the generator of 0 on ρi for all i. Let  = (N, , β) be a stacky fan such that rays {ρ1 , . . . , ρn } span a vector space

Integral Chow Rings of Toric Stacks 5

N ⊗Z R. The toric Deligne–Mumford stack X () associated to  (in the sense of [2]) is constructed as a certain quotient stack which we will review quickly. This quotient method is modeled on the quotient construction by Cox [3]. Indeed, if N is free and 0 ), then the quotient construction in [3] is nothing but that of  corresponds to ( , can X (). If N is free, then by [10, Proposition 5.1] there exists an isomorphism X () ∼ =

X( , 0 ) , where ( , 0 ) corresponds to . For  with N not necessarily free, the toric Deligne–Mumford stack X () is a finite abelian gerbe over some toric stack X( , 0 ) , whose structure is constructed by the technique of taking the stack of roots of an invertible sheaf on X( , 0 ) . Our calculation shows that Stanlely–Reisner rings and toric stacks in our sense have quite a rich relationship that does not arise in the realm of schemes and algebraic spaces. Also, we would like to make things simple and thus we do not consider the case of finite abelian gerbes over toric stacks. See Remark 2.13 for such a case. Quotient presentation. We present toric stacks as quotients, following [3] and [2]. Let L  :=



Spec(k[xρ , ρ ∈ (1)]ρ ∈σ ⊗k k[R]) ⊂ A(1) ×k Spec k[R], / (1) xρ

σ ∈

where R = HomZ (N/( ρ Z · vρ ), Z) ⊂ M. (If rays in  span the vector space N ⊗Z R, then R = 0.) Define a group scheme   m,nρ = 1 for any m ∈ M G (,0 ) := (aρ )ρ∈(1) ∈ G(1) m |ρ aρ     m,n = Spec k xρ±1 , ρ ∈ (1) / ρ xρ ρ − 1 m∈M . There exists a natural action a : L  × G (,0 ) → L  of G (,0 ) on L  that is determined by ⊂ A(1) . Then by [10, Proposition the embedding G (,0 ) → σ ∈ Spec k[xρ , ρ ∈ (1)]ρ ∈σ / (1) xρ 5.1] the quotient stack [L  /G (,0 ) ] is isomorphic to X(,0 ) . Integral Chow groups of a quotient stack. We recall the definition of Chow groups of a quotient stack for the convenience of the reader. Let X be an n-dimensional algebraic space of finite type over a field k and G a linear algebraic group over k that acts on X. We denote by [X/G] the quotient stack. Choose an action of G on an l-dimensional linear space Al such that there exists an open set U ⊂ Al on which G acts freely and whose complement Al − U has codimension more than n − i. The diagonal free action of G on X × U gives rise to the quotient [(X × U )/G] that is an algebraic space. Then we define the ith Chow group Ai ([X/G]) to be Ai+l ([(X × U )/G]) (the i + lth Chow group of [(X × U )/G]). This group is independent of the quotient presentations [X/G] and the representation of G on Al ; see [5, Definition-Proposition 1 and Proposition 16], [14, Section 2.1]. If the dimension of [X/G] is d, then we put Ad−i ([X/G]) = Ai ([X/G]). Notice that even if i is a

6 I. Iwanari

negative integer the Chow group Ai ([X/G]) is not necessarily trivial. The idea behind the definition is a replacement of the stack with a principal Al -bundle which is representable by an algebraic space in sufficiently high codimension. Our computation is based on intersection theory on stacks due to Edidin–Graham, Kresch, and Totaro. For details, we refer to [5], [14], [18].

2 Integral Chow Rings of Toric Stacks In this section, we prove our main theorem. From now on we always assume that the base field k is an algebraically closed field of characteristic zero. Let us fix notation for cycles on toric stacks. If  (resp. (, 0 )) is a fan (resp. a stacky fan), then for a cone δ ∈  we denote by V(δ) (resp. V (δ)) the torus-invariant cycle on the toric variety X (resp. the toric stack X(,0 ) ), which corresponds to δ. If π(,0 ) : X(,0 ) → X denotes a coarse moduli −1 map, then the cycle V (δ) is defined to be the reduced (and irreducible) cycle π(, 0 ) (V(δ))red .

For a ray ρ ∈ , if no confusion seems likely to arise, we may write Dρ (resp. Dρ ) for the torus-invariant divisor V(ρ) (resp. V (ρ)). Given a stacky fan (, 0 ), let us define the Stanley–Reisner ring associated to (, 0 ). Definition 2.1 (Stanley–Reisner ring associated to a stacky fan). Let (, 0 ) be a stacky fan. Consider a polynomial ring Z[Dρ , ρ ∈ (1)]. Let I(,0 ) be the ideal generated by the linear forms ρ∈(1) m, nρ Dρ as m ranges over M. Here, nρ is the generator of 0 on ρ. / . We Let J be the ideal generated by the monomials Dρ1 · · · Dρs such that ρ1 , . . . , ρs ∈ define the Stanley–Reisner ring SR(,0 ) to be Z[Dρ , ρ ∈ (1)]/(I(,0 ) + J ). For a simplicial fan , let us denote by 0can the canonical free-net and set I := I(,0can ) . Then the classical Stanley–Reisner ring SR of  is defined to be Z[Dρ , ρ ∈ (1)]/(I + J ). For the notion of Stanley–Reisner rings associated to fans and polytopes, we refer to [6, Section 5.2], [4, 10.7]. Theorem 2.2. Let (, 0 ) be a stacky fan and X(,0 ) the toric stack associated to (, 0 ). Let A∗ (X(,0 ) ) denote the integral Chow ring of X(,0 ) . Suppose that rays in  span the

Integral Chow Rings of Toric Stacks 7

vector space N ⊗Z R. Then there exists an isomorphism of graded rings ∼

SR(,0 ) → A∗ (X(,0 ) ) which sends Dρ to c1 (OX(,0 ) (Dρ )). If cones σ , τ ∈  span γ ∈ , then we have [V (σ )] · [V (τ )] = [V (γ )]

(2)

in A∗ (X(,0 ) ). The proof of Theorem 2.2 proceeds in several steps. Lemma 2.3. Let be a nonsingular fan. Let SR = Z[Dρ , ρ ∈ (1)]/(I + J ) be the (classical) Stanley–Reisner ring associated to . Then we have the following cases. (i)

In SR , any monomial Dρr11 · · · Dρrnn is equivalent to a linear sum of monomials Dξ1 · · · Dξl , where all the ξ1 , . . . , ξl are distinct and l = r1 + · · · + rn ;

(ii)

If is a fan in N with rk N = d, then SR is zero in degrees higher than d.

Proof. By an argument in the proof of [4, 10.7.1], we can easily show this lemma, but we prove it for the reader’s convenience. We first prove (i). We may assume that all ρ1 , . . . , ρn are distinct. If ρ1 , . . . , ρn ∈ / , then Dρr11 · · · Dρrnn is zero in SR , and thus we assume that ρ1 , . . . , ρn ∈ and r1 ≥ 2. We prove (i) by induction on the number of coincidences in Dρr11 · · · Dρrnn . Since ρ1 , . . . , ρn is a nonsingular cone, thus there exists m ∈ M = HomZ (N, Z) such that m, vρ1 = −1 and m, vρi = 0 for i ≥ 2, where vρi is the first lattice point on ρi . Then Dρ1 = ρ∈{ (1)−ρ1 } m, vρ Dρ in SR . The right-hand side does not involve the terms Dρ2 , . . . , Dρn . Then the number of coincidences in terms ( ρ∈{ (1)−ρ1 } m, vρ Dρ )Dρr11−1 · · · Dρrnn is less than that of Dρr11 · · · Dρrnn . Hence (i) follows. Next we prove (ii). It suffices to show that Dρr11 · · · Dρrnn is zero if r1 + · · · + rn > d. By (i), we may suppose that r1 = · · · = rn = 1 and n > d. If Dρ1 · · · Dρn is not zero in SR , then ρ1 , . . . , ρn is a cone. However, it gives rise to a contradiction since n > d and is nonsingular. This implies (ii).



We will calculate the Picard group of a toric stack in terms of torus-invariant divisors. Proposition 2.4. Let (, 0 ) be a stacky fan and X(,0 ) the toric stack associated to (, 0 ). Let Pic(X(,0 ) ) denote the group of invertible sheaves on X(,0 ),et ´ . Suppose that rays in  span the vector space N ⊗Z R. Then there exists a natural isomorphism of

8 I. Iwanari

groups

:

∼

Z · Dρ /( ρ∈(1) m, nρ · Dρ )m∈M −→ Pic(X(,0 ) ), Dρ → OX(,0 ) (Dρ ),

ρ∈(1)

where

 ρ∈(1)

Z · Dρ is a free abelian group generated by {Dρ }ρ∈(1) .

Proof. We first prove that the Picard group Pic(X(,0 ) ) on X(,0 ) is isomorphic to ⊕ρ∈(1) Z · Dρ /( ρ∈(1) m, nρ · Dρ )m∈M . Put G := G (,0 ) . Note that every invertible sheaf on L  is trivial. Since L  is an open subset of A(1) whose complement A(1) − L  has codimension more than 1, every invertible sheaf on L  is trivial (recall that Pic(A(1) ) = 0). (Here we abuse notation and we write (1) for the cardinal number of (1). In what follows, we often use this notation.) Any invertible sheaf P → [L  /G] amounts to a Gequivariant invertible sheaf on L  , which is given by a character χ : G → Gm . Hence, there exist isomorphisms of groups Pic(X,0 ) ) ∼ = Hom(group k-schemes) (G, Gm ) ∼ =



Z · Dρ /( ρ∈(1) m, nρ · Dρ )m∈M .

ρ∈(1)

Next, we construct  and prove that  is an isomorphism. If C denotes the complement X(,0 ) − Spec k[M], then according to [14, Proposition 2.4.1] there exists the excision sequence Ad−1 (C ) → Ad−1 (X(,0 ) ) → Ad−1 (Spec k[M]) → 0, where Ad−1 (•) denote integral Chow groups; see [14], [5]. Moreover, Ad−1 (Spec k[M]) = 0 and Ad−1 (C ) = Z⊕(irreducible components of C ) = ⊕ρ∈(1) Z · Dρ . Hence, we have a surjective map  : ⊕ρ∈(1) Z · Dρ → Ad−1 (X(,0 ) ) ∼ = Pic(X(,0 ) ) (the last isomorphism follows from [5, Proposition 18]). Since Pic(X( , 0 ) ) is isomorphic to the finitely generated abelian group ⊕ρ∈(1) Z · Dρ /( ρ∈(1) m, nρ · Dρ )m∈M , it is enough to show that for any m ∈ M, ρ∈(1) m, nρ · Dρ maps to zero. Namely, to show our proposition, it suffices to prove that  induces a surjective homomorphism: :



Z · Dρ /( ρ∈(1) m, nρ · Dρ )m∈M −→ Pic(X(,0 ) ), Dρ → c1 (O(Dρ )).

ρ∈(1)

 To see this, consider a free abelian group N˜ := ρ∈(1) Z · eρ and define a homomorphism ˜ be a fan in N˜ R that consists of cones of abelian group h : N˜ → N by eρ → nρ . Let  γ such that γ is a face of the cone ⊕ρ∈(1) R≥0 · eρ and hR (γ ) is a cone in . Then by ˜  ˜ 0can ) → (, 0 ) deter[10, Corollary 3.10] the associated morphism of stacky fans h : (, ˜ →  of fans mines a smooth surjective morphism p : X˜ → X(,0 ) . The morphism h : 

Integral Chow Rings of Toric Stacks 9 p

π(,0 )

corresponds to the composite X˜ → X(,0 ) −→ X . By the construction, the dual ho˜ = HomZ ( M, ˜ Z) sends m to ρ m, nρ · eρ∨ , where eρ∨ is the dual momorphism h∨ : M → M of eρ . Thus, the rational function m ∈ M on X(,0 ) (or X ) defines a torus-invariant divisor p∗ (m) = ρ m, nρ · E ρ on X˜ , where E ρ is the reduced torus-invariant divisor ˜ Since p−1 (Dρ ) = E ρ , thus a rational function m ∈ M corresponding to the ray R≥0 · eρ ∈ . induces ρ m, nρ · Dρ on X(,0 ) .



If two cones σ and τ in  span the cone γ , then V (σ ) and V (τ ) intersect transversally at V (γ ). Indeed, the underlying set of V (σ ) ∩ V (τ ) is equal to V (γ ). Moreover, W is a smooth toric variety and the projection W → [W/G] is a torus-equivariant morphism. The pullbacks of V (σ ) and V (τ ) to W are torus-invariant cycles on W, which intersect transversally. This implies the relation (2). The group scheme G = G (,0 ) has the form Gkm × H , where H is a finite abelian group, and k = #(1) − dim(X(,0 ) ). Set W := L  . Note that W has the form A(1) − Z where Z is a close subscheme. Decompose H into Z/m1 Z × · · · × Z/mr Z. Note that we may view G as a closed subgroup of the maximal algebraic torus in W. Let P be a positive integer. To compute the Chow ring of X(,0 ) , we use the definition of Chow groups due to Edidin–Graham; see Section 1 and [5]. First of all, we shall construct a certain P (k + r)-dimensional representation of G, i.e. an action of G on the affine space V := A P (k+r) such that V has an open set U on which G acts freely and whose complement V \ U has codimension more than P − 1. To this aim, by choosing the primitive mi th root ζi in the base field for 1 ≤ i ≤ r, we embed G = Gkm × Z/m1 Z × · · · × Z/mr Z into the closed group subscheme of Gkm × Grm as follows: G = Gkm × Z/m1 Z × · · · × Z/mr Z → Gkm × Grm ,

  (u, l1 , . . . , lr ) → u, ζ1l1 , . . . , ζrlr ,

where u ∈ Gkm and (l1 , . . . , lr ) ∈ Z/m1 Z × · · · × Z/mr Z. This embedding yields the action of G on an affine space Ak+r . We extend this action to Ak+r × · · · × Ak+r = A P (k+r) diagonally. × A(P −i)(k+r) and U := ∪1≤i≤P Ui . Then the action of G on U is free Set Ui := A(i−1)(k+r) × G(k+r) m and A P (k+r) − U has codimension more than P − 1. Consider the diagonal free action of G on W × U . Then the quotient stack [(W × U )/G] is an algebraic space of finite type and separated over k. Also, [(W × U )/G] is a U -bundle on [W/G] = X(,0 ) . From the definition of Chow groups, we have the following lemma. Lemma 2.5. There exists an isomorphism Ai (X(,0 ) ) = Ai+P (k+r) ([(W × U )/G]) for P > dim(X(,0 ) ) − i = d − i.

10

I. Iwanari

We denote simply by (W × U )/G the algebraic space [(W × U )/G]. The following two lemmas are keys for the proof. Lemma 2.6. The algebraic space (W × U )/G is a smooth toric variety. Proof. Observe that (W × U )/G is smooth. Since W × U → (W × U )/G is smooth surjective and W × U is smooth, thus by [8, Proposition 17.7.7] (or [1, Lemma 5.1.2]) (W × U )/G is smooth. From the construction of the action, every closed point admits a G-stable affine neighborhood, and thus by geometric invariant theory [16, Proposition 1.9] we conclude P (k+r) that (W × U )/G is a scheme. The scheme W × U includes the maximal torus G(1) m × Gm P (k+r) as a dense open subset, and this torus naturally acts on W × U . Since G(1) is m × Gm P (k+r) on W × U descends to (W × U )/G. Thus, the commutative, the action of G(1) m × Gm P (k+r) )/G on itself is naturally extended to (W × U )/G. Recall the fact action of (G(1) m × Gm P (k+r) )/G is that quotients of diagonalizable groups are diagonalizable. Since (G(1) m × Gm P (k+r) )/G is an algebraic split torus. Hence, by the geometric charconnected, (G(1) m × Gm

acterization for toric varieties [13, Theorem 6, p. 24], [17, Theorem 1.5], we conclude that (W × U )/G is a smooth toric variety.



Let N  = Zd+P (k+r) be a lattice and a fan in N  ⊗Z R = NR such that the associated toric variety X is (W × U )/G. Let us denote by (1) the set of rays in corresponding to torus-invariant divisors on (W × U )/G arising from the torus-invariant divisors on W (or equivalently [W/G]). (Notice that (1) also bijectively corresponds to the set of torus-invariant divisors on the toric variety A(1) .) The projection (W × U )/G → [W/G] is a torus-equivariant morphism and induces a bijective map q : (1) → (1) , via the flat pullback of (W × U )/G → [W/G]. Lemma 2.7. (i) (ii)

The fan is a nonsingular fan. Let α1 , . . . , αa be rays in (1) and β1 , . . . , βb rays in (1) − (1) . Let α1 , . . . , αa (resp. β1 , . . . , βb ) be the cone spanned by α1 , . . . , αa (resp. β1 , . . . , βb). Suppose that the cones α1 , . . . , αa and β1 , . . . , βb lie in . Then the cone α1 , . . . , αa , β1 , . . . , βb lies in .

(iii)

If β1 , . . . , β P −1 lie in (1) − (1) , then β1 , . . . , β P −1 span a cone in .

Integral Chow Rings of Toric Stacks 11

Proof. (i) Since (W × U )/G is smooth, thus is a nonsingular fan. (ii) The scheme W × U is also a toric variety in A(1)+P (k+r) , and the projection W × U → (W × U )/G is a torus-equivariant smooth surjective morphism. Let N  = ˜ the fan in NR which represents the toric variety W × U . Here Z(1)+P (k+r) be a lattice and we regard A(1)+P (k+r) as a toric variety determined by a cone spanned by e1 , . . . , e(1)+P (k+r) , ˜ N  = Z(1)+P (k+r) ) → ( , N  = Zd+P (k+r) ) where ei is the ith standard basis for N  . Let φ : ( , be the map of fans which represents the projection W × U → (W × U )/G. Every torusinvariant divisor on W × U descends to that of (W × U )/G, and conversely every torusinvariant divisor on (W × U )/G induces that of W × U via the pullback. This corre∼ ˜ spondence gives rise to a natural bijection (1) = (1). Let α˜ 1 , . . . , α˜ a and β˜1 , . . . , β˜b be ˜ which correspond to α1 , . . . , αa and β1 , . . . , βb, respectively. Note that if X rays in is a simplicial toric variety, then rays ρ1 , . . . , ρn ∈  span a cone in  if and only if ˜ if the intersection V(ρ1 ) ∩ · · · ∩ V(ρn ) is nonempty. Therefore, ei1 , . . . , ein is a cone and only if φR (ei1 , . . . , ein ) is a cone in . Thus, by our assumption, α˜ 1 , . . . , α˜ a and ˜ The cycles ∩1≤i≤a V(α˜ i ) = V(α˜ 1 , . . . , α˜ a ) and ∩1≤i≤b V(β˜i ) = β˜1 , . . . , β˜b are cones in . V(β˜1 , . . . , β˜b ) arise from torus-invariant cycles on W and U , respectively (via pullbacks by natural projections W × U → W and W × U → U ). Therefore, the intersection of V(α1 , . . . , αa ) and V(β1 , . . . , βb ) is nonempty and thus α1 , . . . , αa , β1 , . . . , βb span a cone in . ˜ which correspond to β1 , . . . , β P −1 . These (iii) Let β˜1 , . . . , β˜ P −1 be rays in rays β˜1 , . . . , β˜ P −1 also correspond to the torus-invariant divisors in U . Since U = × A(P −i)(k+r) ) ⊂ A P (k+r) , the intersection V(β˜1 ) ∩ · · · ∩ V(β˜ P −1 ) = ∪1≤i≤P (A(i−1)(k+r) × G(k+r) m V(β˜1 , . . . , β˜ P −1 ) is a nonempty set. Thus, the rays β˜1 , . . . , β˜ P −1 span the cone β˜1 , . . . , β˜ P −1 ∼ ˜ Since (1) ˜ in . = (1) and is nonsingular, φR (β˜1 , . . . , β˜ P −1 ) = β1 , . . . , β P −1 and this 

completes the proof.

From now on, the positive integer P is assumed to be greater than one. According to [7, Proposition 2.1], the group Ad−1+P (k+r) ([(W × U )/G]) = Ad−1+P (k+r) (X ) is generated by torus-invariant divisors V(ρ) as ρ ranges over (1). The group of relations on these divisors (in the Chow group) is generated by the linear forms ρ∈ (1) m, vρ · V(ρ) as m ranges over M  = HomZ (N  , Z). Here, vρ is the first lattice point of ρ. We denote by I the subgroup of ⊕ρ∈ (1) Z · Dρ generated by the above linear forms ρ∈ (1) m, vρ · Dρ . On the other hand, by Proposition 2.4 and Lemma 2.5, there exists a natural isomorphism: Ad−1+P (k+r) ((W × U )/G) = Ad−1 (X(,0 ) ) ∼ =

ρ∈(1)

Z · Dρ /( ρ∈(1) m, nρ · Dρ , m ∈ M).

12

I. Iwanari

Thus, Ad−1+P (k+r) ((W × U )/G) is generated by the torus-invariant divisors arising from the torus-invariant divisors on X(,0 ) ∼ = [W/G]. Therefore, we can choose the element Dξ − ρ∈ (1) aρ · Dρ (aρ ∈ Z)

(3)

in I for each ray ξ in ( (1) − (1) ). The Chow ring A∗ ((W × U )/G) is generated by torusinvariant divisors Dρ as ρ ranges over (1) . Indeed, notice that (W × U )/G is smooth. Thus, by [7, Proposition 2.1], the Chow ring A∗ ((W × U )/G) is generated by torus-invariant divisors Dρ as ρ ranges over (1) because Ad−1+P (k+r) ((W × U )/G) is generated by the torus-invariant divisors arising from the torus-invariant divisors on X(,0 ) ∼ = [W/G]. Let ρ1 , . . . , ρk be rays in . Then ρ1 , . . . , ρk is a cone in  if and only if q(ρ1 ), . . . , q(ρk ) is a cone in . If we regard m ∈ M as a rational function on X , then as observed in the proof of Proposition 2.4 the element m induces the divisor ρ∈(1) m, nρ · Dρ . Thus, the pullback of this divisor to W corresponds to ρ∈(1) m, nρ · Dq(ρ) , and so it lies in I because by [7, Proposition 2.1 (b)] the group of relations on divisors is generated by I . Let ξ˜ : Z[Dρ , ρ ∈ (1)] → Z[Dρ , ρ ∈ (1)] be a ring homomorphism determined by ξ˜ (Dρ ) = Dq(ρ) . Then it induces a surjective homomorphism of graded rings ξ : Z[Dρ , ρ ∈ (1)]/(I(,0 ) + J ) → Z[Dρ , ρ ∈ (1)]/(I + J ), because ξ˜ (I(,0 ) ) ⊂ I and ξ˜ (J ) ⊂ J , respectively. In addition, there exists a natural surjective homomorphism η : Z[Dρ , ρ ∈ (1)]/(I + J ) → A∗ ((W × U )/G), where η(Dρ ) = V(ρ). Next, we show the following lemma. Lemma 2.9. The homomorphism ξ is bijective modulo degree P . Proof. By Lemma 2.5, we have Ad−1 ([W/G]) ∼ = Ad−1+P (k+r) ((W × U )/G), that is, there is the natural isomorphism between the Picard groups of [W/G] and (W × U )/G. Namely, by Proposition 2.4, the natural homomorphism of groups ⊕ρ∈(1) ZDρ /( ρ∈(1) m, nρ · Dρ )m∈M → ⊕ρ∈ (1) ZDρ /I , Dρ → Dq(ρ) is an isomorphism where ⊕ρ∈ (1) ZDρ /I is naturally isomorphic to Ad−1+P (k+r) ((W × U )/G). Thus, ξ is bijective modulo degree 2. Let J (1) (resp. J (2) ) be the ideal generated by the monomials Dρ1 · · · Dρv such that ρ1 , . . . , ρv are in (1) (resp. (1) − (1) ) / . Then by Lemma 2.7 (ii), J is generated by two ideals J (1) and J (2) . and ρ1 , . . . , ρv ∈

Integral Chow Rings of Toric Stacks 13

Moreover, by Lemma 2.7 (iii), J (2) is generated by the monomials whose degree are greater than P − 1. Since I contains (3), thus we have an isomorphism   Z[Dρ , ρ ∈ (1)]/(I + J ) ∼ = Z[Dρ , ρ ∈  (1)]/ I + J (1) ∩ Z[Dρ , ρ ∈  (1)] modulo degree P . A monomial Dρ1 · · · Dρv such that ρ1 , . . . , ρv ∈ (1) and ρ1 , . . . , ρv ∈ / / . Taking these observations into correspond to Dq(ρ1 ) · · · Dq(ρv ) such that q(ρ1 ), . . . , q(ρv ) ∈ account, we conclude that ξ is bijective modulo degree P .



Next, we prove the following lemma. Lemma 2.10. The natural morphism of graded rings η : Z[Dρ , ρ ∈ (1)]/(I + J )→A∗ ((W × U )/G) = A∗ (X ) is an isomorphism. Proof. Let τ be an s-dimensional cone in . Let m be an element in τ ⊥ ∩ M  , where M  = HomZ (N  , Z). Let nρ1 ,...,ρs+1 ,τ be a lattice point in ρ1 , . . . , ρs+1 whose image generates   /Nτ , where Nρ (resp. Nτ ) is a sublattice of N  the one dimensional lattice Nρ 1 ,...,ρs+1 1 ,...,ρs+1

generated by ρ1 , . . . , ρs+1 ∩ N  (resp. τ ∩ N  ). We define e(τ , m) to be m, nρ1 ,...,ρs+1 ,τ Dρ1 · · · Dρs+1 . ρ1 ,...,ρs+1 ⊃τ in

Let us denote by vρ1 ,...,ρs+1 ,τ the first lattice point of the ray of ρ1 , . . . , ρs+1 , which is not contained in τ (it is unique since is a nonsingular fan, in particular simplicial). Since is a nonsingular fan, e(τ , m) is equal to m, vρ1 ,...,ρs+1 ,τ Dρ1 · · · Dρs+1 . ρ1 ,...,ρs+1 ⊃τ in

By Lemma 2.3 (ii), the ring Z[Dρ , ρ ∈ (1)]/(I + J ) has no element whose degree is greater than dim(W × U )/G. Let (L )l be the Z-module generated by elements e(τ , m) such that / . By [7, τ ∈ (l − 1), m ∈ τ ⊥ ∩ M  and the monomials Dρ1 · · · Dρl such that ρ1 , . . . , ρl ∈ Proposition 2.1], the Chow group Ad−l+P (k+r) (X ) is isomorphic to ⎞ ⎛  ⎟ ⎜

ZDρ1 · · · Dρl ⎠ (L )l . ⎝ ρ1 ,...,ρl ∈ (1) ρi =ρ j if i= j

(If ρ1 , . . . , ρl ∈ , then Dρ1 · · · Dρl corresponds to V(ρ1 , . . . , ρl ).) To obtain the lemma, it suffices to prove the following claim.



14

I. Iwanari

Claim 2.10.1. Let (I + J )l be the Z-module generated by the homogeneous elements of the ideal I + J whose degree are equal to l. Suppose that l ≤ dim[(W × U )/G]. Then there exists the canonical isomorphism ⎛ ⎞ ⎛ 



⎜ ⎟ ⎜ ZDρ1 · · · Dρl ⎠ (L )l → ⎝ ⎝ ρ1 ,...,ρl ∈ (1) ρi =ρ j if i= j

⎞



ZDρr11

···

⎟ Dρrss ⎠

(I + J )l .

ρ1 ,...,ρs ∈ (1) r1 ,...,rs ∈Z≥1 , r1 +···+rs =l

Proof. We first show that (I + J )l ⊃ (L )l . To this aim, put τ = ρ1 , . . . , ρl−1 . Then for m in τ ⊥ ∩ M  , we have

⎛ ⎜ e(τ , m) ≡ Dρ1 · · · Dρl−1 ⎝ ⎛ ≡ Dρ1 · · · Dρl−1 ⎝

⎞

⎟ m, vρ Dρ ⎠

ρ∈ (1)−{ρ1 ,...,ρl−1 } such that ρ,τ ∈



⎞

m, vρ Dρ ⎠ .

ρ∈ (1)

Here by ≡, we mean “modulo monomials Dα1 · · · Dαl such that ρ1 , . . . , ρl ∈ / ”. Clearly, if / , then Dα1 · · · Dαl lies in (I + J )l . Hence, (I + J )l ⊃ (L )l . By Lemmas 2.7 α1 , . . . , αl ∈ (i) and 2.3 (ii), we see that the canonical injective map ⎛ ⎞ ⎛ ⎛ 

⎜

⎟ ⎜ ⎜ ZDρ1 · · · Dρl ⎠ ⎝(I + J )l ∩ ⎝ ⎝ ρ1 ,...,ρl ∈ (1) ρi =ρ j if i= j

⎟⎟ ZDρ1 · · · Dρl ⎠⎠

ρ1 ,...,ρl ∈ (1) ρi =ρ j if i= j

⎛ ⎜ →⎝

⎞⎞

⎞





ZDρr11

···

⎟ Dρrss ⎠

(I + J )l

ρ1 ,...,ρs ∈ (1) r1 ,...,rs ∈Z≥1 , r1 +···+rs =l

is bijective. To prove our claim, it suffices to prove that (L )l contains ⎛ ⎞

⎜ ⎟ ZDρ1 · · · Dρl ⎠ . (I + J )l◦ := (I + J )l ∩ ⎝ ρ1 ,...,ρl ∈ (1) ρi =ρ j if i= j

Notice that by [7, Proposition 2.1 (b)] (L )l generates the group of relations on the (dim[(W × U )/G] − l)-dimensional torus-invariant cycles in the Chow group. Then our as/ sertion is clear since (I + J )1 defines rational equivalence relations, and if ρ1 , . . . , ρl ∈ , the intersection V(ρ1 ) ∩ · · · ∩ V(ρl ) is empty. Thus, we have (I + J )l◦ ⊂ (L )l . This implies our claim.



Integral Chow Rings of Toric Stacks 15

By Lemmas 2.9 and 2.10, we obtain the following proposition. Proposition 2.11. The composite η ◦ ξ is bijective modulo degree P . Now we complete the proof of Theorem 2.2. Proof of Theorem 2.2. By Lemma 2.5, we have Ai (X(,0 ) ) = Ai+P (k+r) ((W × U )/G) for P > dim X(,0 ) − i. By Proposition 2.11, the map η ◦ ξ : Z[Dρ , ρ ∈ (1)]/(I(,0 ) + J ) → A∗ ([(W × U )/G]) is bijective modulo degree P . Since P is an arbitrary positive integer, thus we complete 

the proof of Theorem 2.2. Example 2.12.

As mentioned in the Introduction, we present an example of a toric

stack, whose integral Chow ring is nontrivial in degrees higher than the dimension of the stack. (Readers can easily construct such an example.) Let  be a complete fan in NR with rays ρ1 = R≥0 · (2e1 − e2 ), ρ2 = R≥0 · (−e1 + 2e2 ), and ρ3 = R≥0 · (−e1 − e2 ). Then by Theorem 2.2 there exists an isomorphism of graded rings ∼

SR = Z[D1 , D2 , D3 ]/(2D1 − D2 − D3 , −D1 + 2D2 − D3 , D1 D2 D3 ) → A∗ (X(,0can ) ), where Di maps to c1 (O(V (ρi ))) in A1 (X(,0can ) ) for i = 1, 2, 3. Thus, we see that A0 (X(,0can ) ) ∼ = 1 2 ⊕2 k ⊕3 ∼ ∼ ∼ Z, A (X(,0 ) ) = Z ⊕ Z/3Z, A (X(,0 ) ) = Z ⊕ (Z/3Z) and A (X(,0 ) ) = (Z/3Z) for k ≥ can

can

can

3. On the other hand, the Chow group (resp. operational Chow group) A∗ (X ) (resp. A∗ (X )) of the toric variety X are computed as follows: A0 (X ) = Z, A1 (X ) = Z ⊕ Z/3Z, A2 (X ) = Z, Ak (X ) = 0 for k ≥ 3 or k < 0 and Ak (X ) = Z for 0 ≤ k ≤ 2, Ak (X ) = 0 for k ≥ 3, by using [7, Propositions 2.1 and 2.4]. Remark 2.13. Let us mention the work of Jiang and Tseng [11]. After the authors obtained the results in the presented paper, they calculated the integral Chow ring of toric Deligne–Mumford stacks in the sense of [2]. Any toric Deligne–Mumford stack is a finite abelian gerbe over some toric stack in our sense. The gerbe is obtained by the technique of taking the stack of roots of an invertible sheaf on a toric stack (for details, we refer to [10] or [11]). Using structures of gerbes, their calculation was done by reducing it to our main result.

16

I. Iwanari

Acknowledgments I would like to thank Masanori Ishida, Fumiharu Kato, Toshiaki Maeno, and Tadao Oda for their valuable comments.

References [1]

Abramovich, D., A. Corti, and A. Vistoli. “Twisted bundles and admissible covers.” Communications in Algebra 31 (2003): 3547–618.

[2]

Borisov, L., L. Chen, and G. Smith. “Orbifold Chow ring of toric Deligne–Mumford stacks.” Journal of American Mathematical Society 18 (2005): 193–215.

[3]

Cox, D. “The homogeneous coordinate ring of a toric variety.” Journal of Algebraic Geometry 4 (1995): 17–50.

[4]

Danilov, I. V. “The geometry of toric varieties.” Uspekhi Matematicheskikh Nauk 33 (1978): 85–134.

[5]

Edidin, D., and W. Graham. “Equivariant intersection theory.” Inventiones Mathematicae 131 (1998): 595–634.

[6]

Fulton, W. Introduction to Toric Varieties. Annals of Mathematics Studies. Princeton: Princeton University Press, 1993.

[7]

Fulton, W., and B. Sturmfels. “Intersection theory of toric varieties.” Topology 36 (1997):

[8]

335–53. ´ ements Dieudonne, de geom etrie algebrique.” Institut Hautes ´ J., and A. Grothendieck. “El ´ ´ ´ ´ ´ Etudes Scientifiques. Publications Mathematiques 4, 8, 11, 17, 20, 24, 28, 32 (1961–1967). ´

[9]

Iwanari, I. “Logarithmic geometry, minimal free resolutions and toric algebraic stacks.” Preprint available at: http://www.math.kyoto-u.ac.jp/˜iwanari/.

[10]

Iwanari, I. “The category of toric stacks.” Compositio Mathematica 145 (2009): 718–46.

[11]

Jiang, Y., and H.-H. Tseng. “The integral (orbifold) Chow rings of toric Deligne–Mumford stacks.” (2007): preprint arXiv:0707.2972.

[12]

Jurkiewicz, J. “Chow ring of projective nonsingular torus embedding.” Colloquium Mathematicum 43 (1980): 261–70.

[13]

Kempf, G., F. Knudsen, D. Mumford, and B. Saint-Donat. Toroidal Embeddings 1. Lecture Notes in Mathematics 339. Berlin: Springer, 1973.

[14]

Kresch, A. “Cycle groups for Artin stacks.” Inventiones Mathematicae 138 (1999): 495– 536.

[15]

Laumon, G., and L. Moret-Bailly. Champs Algebriques. Ergebnisse der Mathematik und ihrer ´ Grenzgebiete 3, Folge A. Series of Modern Surveys in Mathematics. Berlin: Springer, 2000.

[16]

Mumford, D., J. Fogarty, and F. Kirwan. Geometric Invariant Theory, 3rd ed. Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Folge A. Series of Modern Surveys in Mathematics. Berlin: Springer, 1992.

[17]

Oda, T. Convex Bodies and Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Folge A. Series of Modern Surveys in Mathematics. Berlin: Springer, 1988.

Integral Chow Rings of Toric Stacks 17 [18]

Totaro, B. “The Chow Ring of a Classifying Space.” In Algebraic K-Theory, 249–81. Proceedings of Symposia in Pure Mathematics 67. Providence, RI: American Mathematical Society, 1999.

[19]

Vistoli, A. “Intersection theory on algebraic stacks and their moduli spaces.” Inventiones

[20]

Vistoli, A. “The Chow ring of M2 .” Inventiones Mathematicae 131 (1998): 635–44.

Mathematicae 97 (1989): 613–70.